Modelling extracellular electrical stimulation: III. Derivation and interpretation of neural tissue equations

Enabling Electric Field Model of Microscopically Realistic Brain

BACKGROUND

Modeling brain stimulation at the microscopic scale may reveal new paradigms for various stimulation modalities.

OBJECTIVE

We present the largest map to date of extracellular electric field distributions within a layer L2/L3 mouse primary visual cortex brain sample. This was enabled by the automated analysis of serial section electron microscopy images with improved handling of image defects, covering a volume of 250 × 140 × 90 μm³.

METHODS

The map was obtained by applying a uniform brain stimulation electric field at three different polarizations and accurately computing microscopic field perturbations using the boundary element fast multipole method. We used the map to identify the effect of microscopic field perturbations on the activation thresholds of individual neurons. Previous relevant studies modeled a macroscopically homogeneous cortical volume.

RESULT

Our result shows that the microscopic field perturbations - an 'electric field spatial noise' with a mean value of zero - only modestly influence the macroscopically predicted stimulation field strengths necessary for neuronal activation. The thresholds do not change by more than 10% on average.

CONCLUSION

Under the stated limitations and assumptions of our method, this result essentially justifies the conventional theory of "invisible" neurons embedded in a macroscopic brain model for transcranial magnetic and transcranial electrical stimulation. However, our result is solely sample-specific and is only relevant to this relatively small sample with 396 neurons. It largely neglects the effect of the microcapillary network. Furthermore, we only considered the uniform impressed field and a single-pulse stimulation time course.

Quasistatic approximation in neuromodulation

We define and explain the quasistatic approximation (QSA) as applied to field modeling for electrical and magnetic stimulation. Neuromodulation analysis pipelines include discrete stages, and QSA is applied specifically when calculating the electric and magnetic fields generated in tissues by a given stimulation dose. QSA simplifies the modeling equations to support tractable analysis, enhanced understanding, and computational efficiency. The application of QSA in neuromodulation is based on four underlying assumptions: (A1) no wave propagation or self-induction in tissue, (A2) linear tissue properties, (A3) purely resistive tissue, and (A4) non-dispersive tissue. As a consequence of these assumptions, each tissue is assigned a fixed conductivity, and the simplified equations (e.g. Laplace's equation) are solved for the spatial distribution of the field, which is separated from the field's temporal waveform. Recognizing that electrical tissue properties may be more complex, we explain how QSA can be embedded in parallel or iterative pipelines to model frequency dependence or nonlinearity of conductivity. We survey the history and validity of QSA across specific applications, such as microstimulation, deep brain stimulation, spinal cord stimulation, transcranial electrical stimulation, and transcranial magnetic stimulation. The precise definition and explanation of QSA in neuromodulation are essential for rigor when using QSA models or testing their limits.

A multi-scale simulation of retinal physiology

We present a detailed physiological model of the (human) retina that includes the biochemistry and electrophysiology of phototransduction, neuronal electrical coupling, and the spherical geometry of the eye. The model is a parabolic-elliptic system of partial differential equations based on the mathematical framework of the bi-domain equations, which we have generalized to account for multiple cell-types. We discretize in space with non-uniform finite differences and step through time with a custom adaptive time-stepper that employs a backward differentiation formula and an inexact Newton method. A refinement study confirms the accuracy and efficiency of our numerical method. Numerical simulations using the model compare favorably with experimental findings, such as desensitization to light stimuli and calcium buffering in photoreceptors. Other numerical simulations suggest an interplay between photoreceptor gap junctions and inner segment, but not outer segment, calcium concentration. Applications of this model and simulation include analysis of retinal calcium imaging experiments, the design of electroretinograms, the design of visual prosthetics, and studies of ephaptic coupling within the retina.

A varying-radius cable equation for the modelling of impulse propagation in excitable fibres

In this study, we present a varying-radius cable equation for nerve fibres taking into account the varying diameter along the neuronal segments. Finite element neuronal models utilising the classical (fixed-radius) and varying-radius cable formulations were compared using simple and realistic morphologies under intra- and extracellular electrical stimulation protocols. We found that the use of the classical cable equation to model intracellular neural electrical stimulation exhibited an error of 17% in a passive resistive cable model with abrupt change in radius from 1 to 2 μm, when compared to the known analytical solution and varying-radius cable formulation. This error was observed to increase substantially using more realistic neuron morphologies and branching structures. In the case of extracellular stimulation however, the difference between the classical and varying-radius formulations was less pronounced, but we expect this difference will increase under more complex stimulation paradigms such as high-frequency stimulation. We conclude that for computational neuroscience applications, it is essential to use the varying-radius cable equation for accurate prediction of neuronal responses under electrical stimulation.

Unifying information theory and machine learning in a model of electrode discrimination in cochlear implants

Despite the development and success of cochlear implants over several decades, wide inter-subject variability in speech perception is reported. This suggests that cochlear implant user-dependent factors limit speech perception at the individual level. Clinical studies have demonstrated the importance of the number, placement, and insertion depths of electrodes on speech recognition abilities. However, these do not account for all inter-subject variability and to what extent these factors affect speech recognition abilities has not been studied. In this paper, an information theoretic method and machine learning technique are unified in a model to investigate the extent to which key factors limit cochlear implant electrode discrimination. The framework uses a neural network classifier to predict which electrode is stimulated for a given simulated activation pattern of the auditory nerve, and mutual information is then estimated between the actual stimulated electrode and predicted ones. We also investigate how and to what extent the choices of parameters affect the performance of the model. The advantages of this framework include i) electrode discrimination ability is quantified using information theory, ii) it provides a flexible framework that may be used to investigate the key factors that limit the performance of cochlear implant users, and iii) it provides insights for future modeling studies of other types of neural prostheses.

The Effects of Phase Durations on the Spatial Responses of Retinal Ganglion Cells to Epi- and Sub-Retinal Electrical Stimulation

Retinal prostheses have the potential to restore vision to blind patients that have retinitis pigmentosa or similar hereditary degenerative disorders, by electrically stimulating surviving retinal neurons through implanted electrode arrays. Current retinal prostheses provide limited visual acuity and one challenge is to spatially control neural activation following electrical stimulation. Most of the retinal prostheses are either epi-retinal - in front of the retinal ganglion cell layer, or sub-retinal - behind photoreceptor layer. In this study, we performed calcium imaging of ganglion cells from whole mounted retinas and compared the spread of neural activation between epi-retinal stimulation with a fiber electrode and sub-retinal stimulation with a disk electrode. We investigated the effects of phase durations on the spatial resolution of biphasic stimulation. Our results suggest that with fiber electrode epi-retinal stimulation, the axon bundles activation can lead to significant spread of stimulation, and cannot be avoided simply by changing the phase durations. However, sub-retinal stimulation with very short pulses (phase duration 0.033ms) can effectively confine the activation of retinal ganglion cells.

Determination of the electrical impedance of neural tissue from its microscopic cellular constituents

The electrical properties of neural tissue are important in a range of different applications in biomedical engineering and basic science. These properties are characterized by the electrical admittivity of the tissue, which is the inverse of the specific tissue impedance.

OBJECTIVE

Here we derived analytical expressions for the admittivity of various models of neural tissue from the underlying electrical and morphological properties of the constituent cells.

APPROACH

Three models are considered: parallel bundles of fibers, fibers contained in stacked laminae and fibers crossing each other randomly in all three-dimensional directions.

MAIN RESULTS

An important and novel aspect that emerges from considering the underlying cellular composition of the tissue is that the resulting admittivity has both spatial and temporal frequency dependence, a property not shared with conventional conductivity-based descriptions. The frequency dependence of the admittivity results in non-trivial spatiotemporal filtering of electrical signals in the tissue models. These effects are illustrated by considering the example of pulsatile stimulation with a point source electrode. It is shown how changing temporal parameters of a current pulse, such as pulse duration, alters the spatial profile of the extracellular potential. In a second example, it is shown how the degree of electrical anisotropy can change as a function of the distance from the electrode, despite the underlying structurally homogeneity of the tissue. These effects are discussed in terms of different current pathways through the intra- and extra-cellular spaces, and how these relate to near- and far-field limits for the admittivity (which reduce to descriptions in terms of a simple conductivity).

SIGNIFICANCE

The results highlight the complexity of the electrical properties of neural tissue and provide mathematical methods to model this complexity.

Modified cable equation incorporating transverse polarization of neuronal membranes for accurate coupling of electric fields

OBJECTIVE

We present a theory and computational methods to incorporate transverse polarization of neuronal membranes into the cable equation to account for the secondary electric field generated by the membrane in response to transverse electric fields. The effect of transverse polarization on nonlinear neuronal activation thresholds is quantified and discussed in the context of previous studies using linear membrane models.

APPROACH

The response of neuronal membranes to applied electric fields is derived under two time scales and a unified solution of transverse polarization is given for spherical and cylindrical cell geometries. The solution is incorporated into the cable equation re-derived using an asymptotic model that separates the longitudinal and transverse dimensions. Two numerical methods are proposed to implement the modified cable equation. Several common neural stimulation scenarios are tested using two nonlinear membrane models to compare thresholds of the conventional and modified cable equations.

MAIN RESULTS

The implementations of the modified cable equation incorporating transverse polarization are validated against previous results in the literature. The test cases show that transverse polarization has limited effect on activation thresholds. The transverse field only affects thresholds of unmyelinated axons for short pulses and in low-gradient field distributions, whereas myelinated axons are mostly unaffected.

SIGNIFICANCE

The modified cable equation captures the membrane's behavior on different time scales and models more accurately the coupling between electric fields and neurons. It addresses the limitations of the conventional cable equation and allows sound theoretical interpretations. The implementation provides simple methods that are compatible with current simulation approaches to study the effect of transverse polarization on nonlinear membranes. The minimal influence by transverse polarization on axonal activation thresholds for the nonlinear membrane models indicates that predictions of stronger effects in linear membrane models with a fixed activation threshold are inaccurate. Thus, the conventional cable equation works well for most neuroengineering applications, and the presented modeling approach is well suited to address the exceptions.

Coupling Magnetically Induced Electric Fields to Neurons: Longitudinal and Transverse Activation

We present a theory and computational models to couple the electric field induced by magnetic stimulation to neuronal membranes. Based on the characteristics of magnetically induced electric fields and the modified cable equation that we developed previously, quasipotentials are derived as a simple and accurate approximation for coupling of the electric fields to neurons. The conventional and modified cable equations are used to simulate magnetic stimulation of long peripheral nerves by circular and figure-8 coils. Activation thresholds are obtained over a range of lateral and vertical coil positions for two nonlinear membrane models representing unmyelinated and myelinated straight axons and also for undulating myelinated axons. For unmyelinated straight axons, the thresholds obtained with the modified cable equation are significantly lower due to transverse polarization, and the spatial distributions of thresholds as a function of coil position differ significantly from predictions by the activating function. However, the activation thresholds of unmyelinated axons obtained with either cable equation are very high and beyond the output capabilities of conventional magnetic stimulators. For myelinated axons, threshold values are similar for both cable equations and within the range of magnetic stimulators. Whereas the transverse field contributes negligibly to the activation thresholds of myelinated fibers, axonal undulation can significantly increase or decrease thresholds depending on coil position. The analysis provides a rigorous theoretical foundation and implementation methods for the use of the cable equation to model neuronal response to magnetically induced electric fields. Experimentally observed stimulation with the electric fields perpendicular to the nerve trunk cannot be explained by transverse polarization and is likely due to nerve fiber undulation and other geometrical inhomogeneities.

Hydrogel bioelectronics

Hydrogels have emerged as a promising bioelectronic interfacing material. This review discusses the fundamentals and recent advances in hydrogel bioelectronics.

On the parameters used in finite element modeling of compound peripheral nerves

OBJECTIVE

Computational modeling is an important tool for developing and optimizing implantable neural stimulation devices, but requires accurate electrical and geometrical parameter values to improve predictive value. We quantified the effects of perineurial (resistive sheath around each fascicle) and endoneurial (within each fascicle) parameter values for modeling peripheral nerve stimulation.

APPROACH

We implemented 3D finite element models of compound peripheral nerves and cuff electrodes to quantify activation and block thresholds of model axons. We also implemented a 2D finite element model of a bundle of axons to estimate the bulk transverse endoneurial resistivity; we compared numerical estimates to an analytical solution.

MAIN RESULTS

Since the perineurium is highly resistive, potentials were approximately constant over the cross section of a fascicle, and the perineurium resistivity, longitudinal endoneurial resistivity, and fascicle diameter had important effects on thresholds. Activation thresholds increased up to ~130% for higher perineurium resistivity (~400 versus 2200 Ω m) and by ~35%-250% for lower longitudinal endoneurial resistivity (3.5 versus 0.75 Ω m), with larger increases for smaller diameter axons and for axons in larger fascicles. Further, thresholds increased by ~30%-180% for larger fascicle radii, yielding a larger increase with higher perineurium resistivity. Thresholds were largely insensitive to the transverse endoneurial resistivity, but estimates of the bulk resistivity increased with extracellular resistivity and axonal area fraction; the numerical and analytical estimates were in strong agreement except at high axonal area fractions, where structured axon placements that achieved tighter packing produced electric field inhomogeneities.

SIGNIFICANCE

We performed a systematic investigation of the effects of values and methods for modeling the perineurium and endoneurium on thresholds for neural stimulation and block. These results provide guidance for future modeling studies, including parameter selection, data interpretation, and comparison to experimental results.

Biophysical basis of the linear electrical receptive fields of retinal ganglion cells

Responses of retinal ganglion cells to direct electrical stimulation have been shown experimentally to be well described by linear-nonlinear models. These models rely on the simplifying assumption that retinal ganglion cell responses to stimulation with an array of electrodes are driven by a simple linear weighted sum of stimulus current amplitudes from each electrode, known as the 'electrical receptive field'.

OBJECTIVE

This paper aims to demonstrate the biophysical basis of the linear-nonlinear model and the electrical receptive field to facilitate the development of improved stimulation strategies for retinal implants.

APPROACH

We compare the linear-nonlinear model of subretinal electrical stimulation with a multi-layered, biophysical, volume conductor model of retinal stimulation.

MAIN RESULTS

Our results show that the linear electrical receptive field of the linear-nonlinear model matches the transmembrane currents induced by electrodes (the activating function) at the site of the high-density sodium channel band with only minor discrepancies. The discrepancies are mostly eliminated by including axial current flow originating from adjacent cell compartments. Furthermore, for cells where a single linear electrical receptive field is insufficient, we show that cell responses are likely driven by multiple sites of action potential initiation with multiple distinct receptive fields, each of which can be accurately described by the activating function.

SIGNIFICANCE

This result establishes that the biophysical basis of the electrical receptive field of the linear-nonlinear model is the superposition of transmembrane currents induced by different electrodes at and near the site of action potential initiation. Together with existing experimental support for linear-nonlinear models of electrical stimulation, this provides a firm basis for using this much simplified model to generate more optimal stimulation patterns for retinal implants.

Minimizing activation of overlying axons with epiretinal stimulation: The role of fiber orientation and electrode configuration

Currently, a challenge in electrical stimulation of the retina with a visual prosthesis (bionic eye) is to excite only the cells lying directly under the electrode in the ganglion cell layer, while avoiding excitation of axon bundles that pass over the surface of the retina in the nerve fiber layer. Stimulation of overlying axons results in irregular visual percepts, limiting perceptual efficacy. This research explores how differences in fiber orientation between the nerve fiber layer and ganglion cell layer leads to differences in the electrical activation of the axon initial segment and axons of passage. Approach. Axons of passage of retinal ganglion cells in the nerve fiber layer are characterized by a narrow distribution of fiber orientations, causing highly anisotropic spread of applied current. In contrast, proximal axons in the ganglion cell layer have a wider distribution of orientations. A four-layer computational model of epiretinal extracellular stimulation that captures the effect of neurite orientation in anisotropic tissue has been developed using a volume conductor model known as the cellular composite model. Simulations are conducted to investigate the interaction of neural tissue orientation, stimulating electrode configuration, and stimulation pulse duration and amplitude. Main results. Our model shows that simultaneous stimulation with multiple electrodes aligned with the nerve fiber layer can be used to achieve selective activation of axon initial segments rather than passing fibers. This result can be achieved while reducing required stimulus charge density and with only modest increases in the spread of activation in the ganglion cell layer, and is shown to extend to the general case of arbitrary electrode array positioning and arbitrary target volume. Significance. These results elucidate a strategy for more targeted stimulation of retinal ganglion cells with experimentally-relevant multi-electrode geometries and achievable stimulation requirements.

A Mixed Integer Linear Programming Approach to Electrical Stimulation Optimization Problems

Electrical stimulation optimization is a challenging problem. Even when a single region is targeted for excitation, the problem remains a constrained multi-objective optimization problem. The constrained nature of the problem results from safety concerns while its multi-objectives originate from the requirement that non-targeted regions should remain unaffected. In this paper, we propose a mixed integer linear programming formulation that can successfully address the challenges facing this problem. Moreover, the proposed framework can conclusively check the feasibility of the stimulation goals. This helps researchers to avoid wasting time trying to achieve goals that are impossible under a chosen stimulation setup. The superiority of the proposed framework over alternative methods is demonstrated through simulation examples.

An Evaluation of the Accuracy of Classical Models for Computing the Membrane Potential and Extracellular Potential for Neurons

Two mathematical models are part of the foundation of Computational neurophysiology; (a) the Cable equation is used to compute the membrane potential of neurons, and, (b) volume-conductor theory describes the extracellular potential around neurons. In the standard procedure for computing extracellular potentials, the transmembrane currents are computed by means of (a) and the extracellular potentials are computed using an explicit sum over analytical point-current source solutions as prescribed by volume conductor theory. Both models are extremely useful as they allow huge simplifications of the computational efforts involved in computing extracellular potentials. However, there are more accurate, though computationally very expensive, models available where the potentials inside and outside the neurons are computed simultaneously in a self-consistent scheme. In the present work we explore the accuracy of the classical models (a) and (b) by comparing them to these more accurate schemes. The main assumption of (a) is that the ephaptic current can be ignored in the derivation of the Cable equation. We find, however, for our examples with stylized neurons, that the ephaptic current is comparable in magnitude to other currents involved in the computations, suggesting that it may be significant-at least in parts of the simulation. The magnitude of the error introduced in the membrane potential is several millivolts, and this error also translates into errors in the predicted extracellular potentials. While the error becomes negligible if we assume the extracellular conductivity to be very large, this assumption is, unfortunately, not easy to justify a priori for all situations of interest.

A computational model of orientation-dependent activation of retinal ganglion cells

Currently, a challenge in electrical stimulation for epiretinal prostheses is the avoidance of stimulation of axons of passage in the nerve fiber layer that originate from distant regions of the ganglion cell layer. A computational model of extracellular stimulation that captures the effect of neurite orientation in anisotropic tissue is developed using a modified version of the standard volume conductor model, known as the cellular composite model, embedded in a four layer model of the retina. Simulations are conducted to investigate the interaction of neural tissue orientation, electrode placement, and stimulation pulse duration and amplitude. Using appropriate multiple electrode configurations and higher frequency stimulation, preferential activation of the axon initial segment is shown to be possible for a range of realistic electrode-retina separation distances. These results establish a quantitative relationship between the time-course of stimulation and physical properties of the tissue, such as fiber orientation.

Impedance Spectrum in Cortical Tissue: Implications for Propagation of LFP Signals on the Microscopic Level

Brain research investigating electrical activity within neural tissue is producing an increasing amount of physiological data including local field potentials (LFPs) obtained via extracellular in vivo and in vitro recordings. In order to correctly interpret such electrophysiological data, it is vital to adequately understand the electrical properties of neural tissue itself. An ongoing controversy in the field of neuroscience is whether such frequency-dependent effects bias LFP recordings and affect the proper interpretation of the signal. On macroscopic scales and with large injected currents, previous studies have found various grades of frequency dependence of cortical tissue, ranging from negligible to strong, within the frequency band typically considered relevant for neuroscience (less than a few thousand hertz). Here, we performed a detailed investigation of the frequency dependence of the conductivity within cortical tissue at microscopic distances using small current amplitudes within the typical (neuro)physiological micrometer and sub-nanoampere range. We investigated the propagation of LFPs, induced by extracellular electrical current injections via patch-pipettes, in acute rat brain slice preparations containing the somatosensory cortex in vitro using multielectrode arrays. Based on our data, we determined the cortical tissue conductivity over a 100-fold increase in signal frequency (5-500 Hz). Our results imply at most very weak frequency-dependent effects within the frequency range of physiological LFPs. Using biophysical modeling, we estimated the impact of different putative impedance spectra. Our results indicate that frequency dependencies of the order measured here and in most other studies have negligible impact on the typical analysis and modeling of LFP signals from extracellular brain recordings.

On the computation of a retina resistivity profile for applications in multi-scale modeling of electrical stimulation and absorption

This study proposes a methodology for computationally estimating resistive properties of tissue in multi-scale computational models, used for studying the interaction of electromagnetic fields with neural tissue, with applications to both dosimetry and neuroprosthetics. Traditionally, models at bulk tissue- and cellular-level scales are solved independently, linking resulting voltage from existing resistive tissue-scale models as extracellular sources to cellular models. This allows for solving the effects that external electric fields have on cellular activity. There are two major limitations to this approach: first, the resistive properties of the tissue need to be chosen, of which there are contradicting measurements in literature; second, the measurements of resistivity themselves may be inaccurate, leading to the mentioned contradicting results found across different studies. Our proposed methodology allows for constructing computed resistivity profiles using knowledge of only the neural morphology within the multi-scale model, resulting in a practical implementation of the effective medium theory; this bypasses concerns regarding the choice of resistive properties and accuracy of measurement setups. A multi-scale model of retina is constructed with an external electrode to serve as a test bench for analyzing existing and resulting resistivity profiles, and validation is presented through the reconstruction of a published resistivity profile of retina tissue. Results include a computed resistivity profile of retina tissue for use with a retina multi-scale model used to analyze effects of external electric fields on neural activity.

Brain Neuromodulation Techniques

The modulation of brain function via the application of weak direct current was first observed directly in the early 19th century. In the past 3 decades, transcranial magnetic stimulation and deep brain stimulation have undergone clinical translation, offering alternatives to pharmacological treatment of neurological and neuropsychiatric disorders. Further development of novel neuromodulation techniques employing ultrasound, micro-scale magnetic fields and optogenetics is being propelled by a rapidly improving understanding of the clinical and experimental applications of artificially stimulating or depressing brain activity in human health and disease. With the current rapid growth in neuromodulation technologies and applications, it is timely to review the genesis of the field and the current state of the art in this area.

A Simple and Accurate Model to Predict Responses to Multi-electrode Stimulation in the Retina

Implantable electrode arrays are widely used in therapeutic stimulation of the nervous system (e.g. cochlear, retinal, and cortical implants). Currently, most neural prostheses use serial stimulation (i.e. one electrode at a time) despite this severely limiting the repertoire of stimuli that can be applied. Methods to reliably predict the outcome of multi-electrode stimulation have not been available. Here, we demonstrate that a linear-nonlinear model accurately predicts neural responses to arbitrary patterns of stimulation using in vitro recordings from single retinal ganglion cells (RGCs) stimulated with a subretinal multi-electrode array. In the model, the stimulus is projected onto a low-dimensional subspace and then undergoes a nonlinear transformation to produce an estimate of spiking probability. The low-dimensional subspace is estimated using principal components analysis, which gives the neuron’s electrical receptive field (ERF), i.e. the electrodes to which the neuron is most sensitive. Our model suggests that stimulation proportional to the ERF yields a higher efficacy given a fixed amount of power when compared to equal amplitude stimulation on up to three electrodes. We find that the model captures the responses of all the cells recorded in the study, suggesting that it will generalize to most cell types in the retina. The model is computationally efficient to evaluate and, therefore, appropriate for future real-time applications including stimulation strategies that make use of recorded neural activity to improve the stimulation strategy.

Implantable multi-electrode arrays (MEAs) are used to record neurological signals and stimulate the nervous system to restore lost function (e.g. cochlear implants). MEAs that can combine both sensing and stimulation will revolutionize the development of the next generation of devices. Simple models that can accurately characterize neural responses to electrical stimulation are necessary for the development of future neuroprostheses controlled by neural feedback. We demonstrate a model that accurately predicts neural responses to concurrent stimulation across multiple electrodes. The model is simple to evaluate, making it an appropriate model for use with neural feedback. The methods described are applicable to a wide range of neural prostheses, thus greatly assisting future device development.

Modelling extracellular electrical stimulation: part 4. Effect of the cellular composition of neural tissue on its spatio-temporal filtering properties

OBJECTIVE

The objective of this paper is to present a concrete application of the cellular composite model for calculating the membrane potential, described in an accompanying paper.

APPROACH

A composite model that is used to determine the membrane potential for both longitudinal and transverse modes of stimulation is demonstrated.

MAIN RESULTS

Two extreme limits of the model, near-field and far-field for an electrode close to or distant from a neuron, respectively, are derived in this paper. Results for typical neural tissue are compared using the composite, near-field and far-field models as well as the standard isotropic volume conductor model. The self-consistency of the composite model, its spatial profile response and the extracellular potential time behaviour are presented. The magnitudes of the longitudinal and transverse components for different values of electrode-neurite separations are compared.

SIGNIFICANCE

The unique features of the composite model and its simplified versions can be used to accurately estimate the spatio-temporal response of neural tissue to extracellular electrical stimulation.